Algebraic Conditions for a Centre or a Focus in Some Simple Systems of Arbitrary Degree

Necessary and sufficient conditions are investigated, in the form of explicit algebraic formulas, for the origin to be a centre, an unstable focus, or a stable focus of a homogeneous polynomial planar vector field whose degree is arbitrary. This provides a unified treatment of the problem. The role of these conditions is exemplified in detail in the linear, cubic, and quintic cases. Similar considerations are provided for systems which possess homogeneous non-linearities and for which the origin is a uniformly isochronous centre